Firstly, we only need to think about the upper number in the meter. The lower number in just tells us which note is counted as one beat.
Thus if we have a 4/3 meter, or a 4/4 meter - they are notationally exactly the same, apart that we need to use quarter notes instead of third-notes. But, I never saw non-dyadic notes in my live.
For the lower number there are two things to consider:
- Of course it is possible to rewrite music of any time signature into any other time signature. You just need tuplets or some complicated dotted notes or even slurs. But usually, as you said, the rhythm will get messy to read.
E.g. To write a waltz in a 4/4 meter, you use a 3:4 tuplet.
Before 1930 (roughly), composers did not like to change time signatures in a piece. Thus, if they had a metre change, they used tuplets and dotted notes.
- In western music, the time signature not only determines how many notes are in each bar, but also the importance of each and every beat. E.g. if you have a 4/4 time signature, then the quaters are: heavy - light - medium - very light.
Thus, if you rewrite your ryhthms, you also have to indicate the stress of every note.
EDIT (since the question changed)
I never saw a non-dyadic note printed somewhere. It seems to me, that non-dyadic notes are used today to make a meter change, while maintaining the tempo.
In former times this was indicated usually by writing e.g.
"quarter = dotted quarter" above the bar-line where the meter change occurs.
Another way was by writing "L'istesso tempo".
One example where this occurs is Schubert, 3 Klavierstücke, #2.
In both of these cases, one can stick to dyadic meters, which makes the rhythm even easier to read then when using non-dyadic meters in my opinion.